## Pricing Down-and-out call options

This is a knock out barrier option. The option “dies” when the underlying’s price crosses a certain barrier price, H. The barrier lies below the underlying’s price at inception, hence the “down” in the title above. The adjustments that will be made to the basic options pricing procedure will be as follows:

**Option Pricing Step2**: The terminal payoffs will be

- if S>H, max{0,S
_{T}-K} - if S<=H, 0

**Option Pricing Step3**: The intrinsic values at non-terminal nodes will be

- if S>H, C, where C = value of the option at the node as defined earlier
- if S<=H, 0

For n=2 this is given below:

Column | A | B | C |

Row | S_{T} | C_{T} | T-1?t |

1 | =u^{2}S_{0} | = if[A1<=H, 0, max(A1-K,0)] | =if[A1<=H, 0, exp(-r?t)*{p*B1+(1-p)*B2}] |

2 | =A1*d | = if[A2<=H, 0, max(A2-K,0)] | =if[A2<=H, 0, exp(-r?t)*{p*B1+(1-p)*B3}] |

3 | =A2*d | = if[A3<=H, 0, max(A3-K,0)] | =if[A3<=H, 0, exp(-r?t)*{p*B2+(1-p)*B4}] |

4 | =A3*d | = if[A4<=H, 0, max(A4-K,0)] | =if[A4<=H, 0, exp(-r?t)*{p*B3+(1-p)*B5}] |

5 | =A4*d | = if[A5<=H, 0, max(A5-K,0)] | =if[A5<=H, 0, exp(-r?t)*{p*B4+(1-p)*B5}] |

Figure 92: Formulas for spreadsheet implementation of binomial tree for down & out call options

The adjusted formulas in this column will then be copied to D and later columns depending on the number of steps employed.

For the following parameters the numerical example is given below:

n | 4 | T | 0.1 | x | 0.9988 |

r | 0.05 | ?t | 0.025 | K | 50 |

? | 0.3 | d | 0.9537 | q | 0.4987 |

u | 1.0486 | p | 0.5013 | S_{0} | 47 |

The Barrier is set at H=45.

Column | A | B | C | D | E | F |

Row | S_{T} | C_{T} | T-1?t | T-2?t | T-3?t | T-4?t |

1 | 56.820 | 6.820 | 5.500 | 4.871 | 4.330 | 4.014 |

2 | 54.187 | 4.187 | 4.250 | 3.798 | 3.707 | 3.376 |

3 | 51.677 | 1.677 | 2.097 | 2.546 | 2.425 | 2.595 |

4 | 49.283 | – | 0.840 | 1.050 | 1.484 | 1.476 |

5 | 47.000 | – | – | 0.420 | 0.526 | 0.743 |

6 | 44.823 | – | – | – | – | – |

7 | 42.746 | – | – | – | – | – |

8 | 40.766 | – | – | – | – | – |

9 | 38.877 | – | – | – | – | – |

Figure 93: Spreadsheet for down & out call option example

The price of the option is 0.743.

The binomial method tends to overstate the value of the option and therefore simply increasing the number of steps does not necessarily guarantee a more accurate value. In fact the error could be greater. This is particularly true if the barrier is not equal to any of the prices on the nodes of the tree. This means that tree implicitly underestimates the chance of a knock out thus making the barrier option more valuable.

An appropriate n is found by solving for the following formula:

where m is some integer. The result is rounded down to the nearest integer.

For the above parameters good values of n may be as follows:

M | 1 | 2 | 3 | 4 | 5 | 6 |

good values of n | 4 | 19 | 42 | 76 | 118 | 171 |

Figure 94: Good values of ‘n’ for Barrier Options

and so on.

If you would rather use Monte Carlo Simulation for pricing Knock out and Sudden death options you would find the following free posts useful.